Download Scattered Sentences have Few Separable Randomizations

arXiv:1704.04981v1 [math.LO] 17 Apr 2017
SCATTERED SENTENCES HAVE FEW SEPARABLE
RANDOMIZATIONS
URI ANDREWS, ISAAC GOLDBRING, SHERWOOD HACHTMAN, H. JEROME
KEISLER, AND DAVID MARKER
Abstract. In the paper Randomizations of Scattered Sentences, Keisler
showed that if Martin’s axiom for aleph one holds, then every scattered
sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is
“yes”. It follows that the absolute Vaught conjecture holds if and only
if every Lω1 ω -sentence with few separable randomizations has countably
many countable models.
1. Introduction
This note answers a question posed in the paper [K2], and results from
a discussion following a lecture by Keisler at the Midwest Model Theory
meeting in Chicago on April 5, 2016.
Fix a countable first order signature L. A sentence ϕ of the infinitary
logic Lω1 ω is scattered if there is no countable fragment LA of Lω1 ω such
that ϕ has a perfect set of countable models that are not LA -equivalent.
Scattered sentences were introduced by Morley [M], motivated by Vaught’s
conjecture. The absolute form of Vaught’s conjecture for an Lω1 ω -sentence
ϕ says that if ϕ is scattered then ϕ has countably many (non-isomorphic)
countable models 1.
In continuous logic, the pure randomization theory P R (from [BK]) is a
theory whose signature LR has a sort K for random elements and a sort E for
events. For each formula θ(·) of L with n free variables, LR has a function
symbol Jθ(·)K of sort Kn → E for the event at which θ(·) is true. LR also
has Boolean operations ⊔, ⊓, ¬ in the event sort, a predicate µ from events
to [0, 1], and distance predicates dK , dE for each sort. The set of axioms for
P R is recursive in L. It insures that the functions Jθ(·)K respect validity,
connectives, and quantifiers, that each event is equal to the set where some
pair of random elements agree, and that µ is an atomless probability measure
on the events. There are also axioms that define dK and dE in the natural
way.
The work of Andrews was partially supported by NSF grant DMS-1600228. The work
of Goldbring was partially supported by NSF CAREER grant DMS-1708802.
1Here, countable means of cardinality at most ℵ .
0
1
SCATTERED SENTENCES HAVE FEW SEPARABLE RANDOMIZATIONS
2
Pre-models of P R are called randomizations, and models of P R are called
complete randomizations. In Theorem 5.1 of [K2] (stated as Fact 2.4 below),
it is shown that in a complete separable randomization, there is a unique
mapping J·K from Lω1 ω -sentences to events that respects validity, countable
connectives, and quantifiers. A separable randomization of an Lω1 ω -sentence
ϕ is a separable randomization whose completion satisfies µ(JϕK) = 1. Intuitively, in a separable randomization of ϕ, a random element is obtained
by randomly picking an element of a random countable model of ϕ, with
respect to some underlying probability space. An especially simple kind
of randomization of ϕ, called a basic randomization, has random elements
picked from some fixed countable family of countable models of ϕ, with
the underlying probability space being the Lebesgue measure on the unit
interval. ϕ is said to have few separable randomizations if every complete
separable randomization of ϕ is isomorphic to a basic randomization.
The main results of [K2] are: If an Lω1 ω -sentence ϕ has countably many
countable models, then ϕ has few separable randomizations. If ϕ has few
separable randomizations, then ϕ is scattered. If Martin’s axiom for ℵ1
holds and ϕ is scattered, then ϕ has few separable randomizations. [K2]
asks whether the conclusion of this last result can be proved in ZFC. Here
we will show that the answer to that question is “yes”. The idea will be
to use the Shoenfield absoluteness theorem to eliminate the use of Martin’s
axiom.
The results in the preceding paragraph show that being scattered is equivalent to having few separable randomizations. The absolute Vaught conjecture for ϕ says that if ϕ is scattered then ϕ has countably many countable
models. Thus the absolute Vaught conjecture is equivalent to the property
that having few separable randomizations implies having countably many
countable models.
2. Background
We refer to [BBHU] for background in continuous logic, [J] for background
on absoluteness and Martin’s axiom, and [K1] for background on Lω1 ω . We
assume throughout that ϕ is an Lω1 ω -sentence that implies (∃x)(∃y)x 6= y.
We will not need the formal statement of the axioms of P R , or the formal
definition of Jψ(·)K for Lω1 ω -formulas ψ(·). In this section we will state the
definitions and results from [K2] that we will need.
Given two pre-structures N and P with signature LR , an isomorphism
h : N → P is a mapping from N into P such that h preserves the truth
values of all formulas of LR , and every element of P is at distance zero
from some element of h(N ). We call P a reduction of N if P is obtained
from N by identifying elements at distance zero, and call P a completion
of N if P is a structure obtained from a reduction of N by completing the
metrics. Up to isomorphism, every pre-structure has a unique reduction and
completion. The mapping that identifies elements at distance zero is called
SCATTERED SENTENCES HAVE FEW SEPARABLE RANDOMIZATIONS
3
the reduction mapping, and is an isomorphism from a pre-structure onto
its reduction.
The axioms of P R have the following consequences:
dK (f , g) = µ(Jf 6= gK),
dE (A, B) = µ(A△B).
µ(J(∃x)(∃y)x 6= yK) = 1.
By the latter, every separable randomization is a separable randomization
of (∃x)(∃y)x 6= y. Since P R has axioms saying that the functions Jθ(·)K for
first order θ respect connectives, and that every event is equal to Ja = bK
for some a, b, it follows that:
Fact 2.1. Suppose N = (K, B) and N ′ = (K′ , B ′ ) are models of P R , h maps
K onto K′ , and
′
N |= µ(Jθ(~a)K) ≥ r ⇔ N ′ |= µN (Jθ(h~a)K) ≥ r
for all first order θ, tuples ~a in K, and rational r. Then h can be extended
to a unique isomorphism from N onto N ′ .
The simplest examples of randomizations are the Borel randomizations,
defined as follows. Let L be the family of Borel subsets of [0, 1) and λ be
the restriction of Lebesgue measure to L.
Definition 2.2. The Borel randomization of a model M |= (∃x)(∃y)x 6= y
is the structure (ML , L) of sort LR where ML is the set of all functions
f : [0, 1) → M with countable range such that {t | f (t) = a} ∈ L for each
a ∈ M , L has the usual Boolean operations, µ is interpreted by λ, and
~ )K = {t | M |= θ(f
~ (t))}.
Jθ(f
A basic randomization of ϕ is formed by “gluing together” countably
many Borel randomizations of countable models of ϕ.
Definition 2.3. Suppose that
S
• [0, 1) = n Bn is a partition of [0, 1) into countably many Borel sets
of positive measure;
• for
n, Mn is a countable model of ϕ;
S
Q each
B
n
• n Mn is the set of all functions f : [0, 1) → n Mn such that for
all n,
(∀t ∈ Bn )f (t) ∈ Mn and (∀a ∈ Mn ){t ∈ Bn | f (t) = a} ∈ L;
Q
n
• ( n MB
n , L) has the usual Boolean operations, µ is interpreted by
λ, and the Jθ(·)K functions are
[
~ )K = {t ∈ Bn | Mn |= θ(f
~ (t))}.
Jθ(f
n
Q
n
( n MB
n , L) is called a basic randomization of ϕ.
Fact 2.4. (Theorem 5.1 in [K2]) Let P = (K, E) be a complete separable
randomization, and let Ψn be the class of Lω1 ω formulas
S with n free variables.
There is a unique family of functions Jψ(·)KP , ψ ∈ n Ψn , such that:
SCATTERED SENTENCES HAVE FEW SEPARABLE RANDOMIZATIONS
4
(i) When ψ ∈ Ψn , Jψ(·)KP : Kn → E.
(ii) When ψ is a first order formula, Jψ(·)KP is the usual event function
for the structure P.
~ )KP = ¬Jψ(f
~ )KP .
(iii) J¬ψ(f
~ )KP = Jψ1 (f
~ )KP ⊔ Jψ2 (f
~ )KP .
(iv) J(ψ1 ∨ ψ2 )(f
W
P
P
~
~
(v) J k ψk (f )K = supk Jψk (f )K .
~ )KP = sup
~ P
(vi) J(∃u)θ(u, f
g∈K Jθ(g, f )K .
Moreover, for each ψ ∈ Ψn , the function Jψ(·)KP is Lipschitz continuous
~,h
~ ∈ Kn we have
with bound one, that is, for any pair of n-tuples f
X
~ )KP , Jψ(h)K
~ P) ≤
dE (Jψ(f
dK (f , hm ).
m
m<n
Definition 2.5. Let N be a separable randomization with completion P,
and ϕ be an Lω1 ω -sentence. We write
µN (JϕK) = µP (JϕK) = µ(JϕKP ).
If µN (JϕK) = 1, we say that N is a randomization of ϕ.
We say that ϕ has few separable randomizations if every complete separable randomization of ϕ is isomorphic to a basic randomization of ϕ.
Fact 2.6. ([K2], Lemma 4.3 and Theorem 4.6.) Every basic randomization
of ϕ is isomorphic to its reduction, which is a complete separable randomization of ϕ (and thus a model of P R ).
Q
B
Fact 2.7. (Lemma 9.4 in [K2]) Let ( j∈J Mj j , L) be a basic randomization.
For each j ∈ J, let δj be a Scott sentence of Mj . Then for each complete
separable randomization P of ϕ, the following are equivalent.
Q
B
• P is isomorphic to ( j∈J Mj j , L).
• µP (Jδn K) = λ(Bj ) for each j ∈ J.
Fact 2.8. (Lemma 9.5 in [K2]) ϕ has few separable randomizations if and
only if for every complete separable randomization (or every countable randomization) N of ϕ there is a Scott sentence δ such that µN (JδK) > 0.
Fact 2.9. (Theorem 10.1 in [K2]). If ϕ has few separable randomizations,
then ϕ is scattered.
Fact 2.10. (Theorem 10.3 in [K2]). Assume that Lebesgue measure is ℵ1 additive (e.g. assume that MA(ℵ1 ) holds). Then every scattered sentence
has few separable randomizations.
Question 11.4 in [K2] asks whether or not the conclusion of Fact 2.10 can
be proved in ZFC.
3. The Main Result
We will prove the following theorem, which answers Question 11.4 in [K2]
affirmatively.
SCATTERED SENTENCES HAVE FEW SEPARABLE RANDOMIZATIONS
5
Theorem 3.1. Every scattered sentence has few separable randomizations.
Fact 2.9 and Theorem 3.1 give us the following two corollaries.
Corollary 3.2. A sentence of Lω1 ω is scattered if and only if it has few
separable randomizations.
Corollary 3.3. For each Lω1 ω -sentence ϕ, the following are equivalent.
(i) The absolute Vaught conjecture for ϕ holds.
(ii) If ϕ has few separable randomizations, then ϕ has countably many
countable models.
Note that each countable pre-structure N = (K, B) in the signature LR
can be coded in a natural way by a first order structure with universe N and
a countable signature indexed by N. In particular, the function µ : B → [0, 1]
can be coded by the set of (e, m, n) ∈ N3 such that e codes an event E and
m/n ≤ µ(E).
Let A be the set of subsets of [0, 1) that are finite unions of intervals
with rational endpoints. Given a countable model M of (∃x)(∃y)x 6= y with
countable signature L, let MA be the set of functions f : [0, 1) → M with
f be the completion
finite range such that for each a ∈ M, f −1 (a) ∈ A. Let M
A
f is isomorphic to the Borel randomization (ML , L) of M. A,
of (M , A). M
M, and MA are countable and can be coded in the natural way by subsets
of N.
Lemma 3.4. Let N = (K, B) be a countable randomization with a coding.
Then the statement (S) below is equivalent (in ZFC) to a Σ11 formula with
parameter N .
(S) There exists a Scott sentence δ such that µN (JδK) > 0.
Proof. For each event C in the completion of N such that µ(C) > 0, let µ|C
be the conditional measure such that
(µ|C)(E) = µ(E ⊓ C)/µ(C),
and let N |C be the completion of the pre-structure obtained from N by
replacing µ by µ|C. We first show that (S) is equivalent to the following
statement.
(S’) There exists a countable model M of (∃x)(∃y)x 6= y and an event C
f
in the completion of N such that µ(C) > 0 and N |C ∼
= M.
Assume (S). Let δ be a Scott sentence δ such that µN (JδK) > 0. Let
C = JδK, which is an event of positive measure in the completion of N .
Then µN |C (JδK) = 1, so N |C is a separable randomization of δ. Let M be a
f so (S’) holds.
countable model of δ. By Fact 2.7, we have N |C ∼
= M,
Now assume (S’). By Scott’s theorem, M has a Scott sentence δ. Then
f
by Fact 2.7, µM (JδK) = 1, so
1 = µN |C (JδK) = µN (C ⊓ JδK)/µN (C).
SCATTERED SENTENCES HAVE FEW SEPARABLE RANDOMIZATIONS
6
Hence
µN (JδK) ≥ µN (C ⊓ JδK) = µN (C) > 0,
so (S) holds.
We now show that (S’) is equivalent to the following statement.
(S”) There exists a countable coded structure M with at least 2 elements,
a sequence B : N → B, and double sequences α : N × N → MA ,
β : N × N → K such that
(a) B is Cauchy convergent in dE , and limn→∞ µ(Bn ) > 0.
(b) For each m ∈ N, hαm,n | n ∈ Ni and hβm,n | n ∈ Ni are Cauchy
convergent in dK .
(c) For each x ∈ MA , there exists mx ∈ N such that αmx ,n = x for
all n ∈ N, and for each y ∈ K, there exists my ∈ N such that
βmy ,n = y for all n ∈ N.
(d) For each L-formula ψ(v1 , . . . , vk ),
f
lim µM (Jψ(α1,n , . . . , αk,n )K) = lim µN (Jψ(β1,n , . . . , βk,n )K ⊓ Bn )/µN (Bn ).
n→∞
n→∞
In (S”), N and M are coded structures, so (S”) is clearly Σ11 with parameter N .
The functions Jψ(·)K are uniformly continuous in each model of P R .
Whenever (a) and (b) hold, for each m, n ∈ N the reduction maps send
′′
f and βm,n to an element βm,n
αm,n to an element α′′m,n of M,
of N |C, and
′
′′
′
′′
f
the limits αm = limn→∞ αm,n in M and βm = limn→∞ βm,n in N |C exist.
Therefore, (a) and (b) imply that for each L-formula ψ(v1 , . . . , vk ),
(3.1)
f
f
µM (Jψ(α′1 , . . . , α′k )K) = lim µM (Jψ(α1,n , . . . , αk,n )K)
n→∞
and
(3.2) µN |C (Jψ(β1′ , . . . , βk′ )K) = lim µN (Jψ(β1,n , . . . , βk,n )K ⊓ Bn )/µN (Bn ).
n→∞
We next assume that (S’) holds for some M and C, and prove (S”). We
may take M to be a coded structure, and let h be an isomorphism from N |C
f We may choose mappings α′ from N into M
f and β ′ from N into N |C
to M.
such that range(α′ ), range(β ′ ) contain the images of MA and K under the
reduction maps, and α′n = h(βn′ ) for each n ∈ N. Then for each L-formula
ψ(v1 , . . . , vk ),
(3.3)
f
µM (Jψ(α′1 , . . . , α′k )K) = µN |C (Jψ(β1′ , . . . , βk′ )K).
One can choose a sequence B : N → B, and double sequences α : N × N →
MA , β : N × N → K such that (c) holds, the reduction of Bn converges to
C, and for each m ∈ N the reductions of αm,n and βm,n converge to α′m and
′ respectively. Then conditions (a) and (b) hold, so (3.1) and (3.2) hold
βm
for each L-formula ψ(v1 , . . . , vk ), By (3.3), condition (d) holds, and hence
(S”) holds.
SCATTERED SENTENCES HAVE FEW SEPARABLE RANDOMIZATIONS
7
Finally, we assume (S”) and prove (S’). Let C = limn→∞ Bn in the completion of B. Since (a) and (b) hold, (3.1) and (3.2) hold for each L-formula
ψ(v1 , . . . , vk ). Then by (d), (3.3) holds for every ψ. By (c), range(α′ ) ⊇ MA
f and
and range(β ′ ) ⊇ K. Therefore range(α′ ) is dense in the K-sort of M,
′
f
range(β ) is dense in the K-sort of N |C. Hence every element of M of sort
K is equal to limk→∞ α′mk for some sequence (m0 , m1 , . . .) ∈ NN , and similarly for N |C and β ′ . Since dK (a, b) = µ(Ja 6= bK) in any model of P R ,
′
f if and only if limk→∞ βm
exists in N |C. Whenlimk→∞ α′mk exists in M
k
′
′
f let h(limk→∞ α ) = limk→∞ β ′ . Then h
ever limk→∞ αmk exists in M,
mk
mk
f onto the K-sort of N |C. Since (3.3) holds and the
maps the K-sort of M
f and N |C,
functions Jψ(·)K are uniformly continuous in M
f
µM (Jψ(~a)K) = µN |C (Jψ(h~a)K)
f Therefore by Fact 2.1, h
for each L-formula ψ and tuple ~a of sort K in M.
f
can be extended to an isomorphism from M onto N |C. This proves (S’). By a transitive model of a set of sentences Z we mean a transitive set V
such that (V, ∈) |= Z. It is well known that there is a finite subset ZFC0
of the set of axioms of ZFC such that the Shoenfield absoluteness theorem
holds for all transitive models of ZFC0 . Assume hereafter that ZFC0 is a
finite subset of ZFC with that property, and also that ZFC0 implies every
result stated in Section 2, Lemma 3.4 above, and every consequence of ZFC
that is used in the proofs of Lemmas 3.5 and 3.6 below.
Lemma 3.5. Let V, V [G] be transitive models of ZFC0 such that the signature L is in V , and V ⊆ V [G]. Suppose that in V it is true that ϕ is an
Lω1 ω -sentence and N = (K, B) is a countable randomization. Then in V [G]
it is also true that ϕ is an Lω1 ω -sentence and N = (K, B) is a countable
randomization, and µN (JϕK) has the same value in V as in V [G]. Hence
V |= N is a countable randomization of ϕ
if and only if
V [G] |= N is a countable randomization of ϕ.
Proof. It is easily proved using induction on the complexity of formulas that
V [G] |= ϕ is a an Lω1 ω -sentence.
Since the set of axioms of P R is recursive in L, the property of being a
countable randomization is Σ1 , and hence
V [G] |= N is a countable randomization.
Let P be the completion of N in V , and Q be the completion of N in V [G].
In V [G], P is a separable randomization that is not necessarily complete,
and Q is the completion of N and also the completion of P. For each Lω1 ω formula ψ(·) in V , let Jψ(·)KP be the function obtained by applying Fact 2.4
SCATTERED SENTENCES HAVE FEW SEPARABLE RANDOMIZATIONS
8
to P in V , and let Jψ(·)KQ be the function obtained by applying Fact 2.4
to Q in V [G]. Using Conditions (i)–(vi) of Fact 2.4, we show by induction
~ in the
on complexity that for every Lω1 ω -formula ψ(·) in V and tuple f
P
Q
~ )K = Jψ(f
~ )K . The base step for first order formulas
reduction of K, Jψ(f
and the steps for negation and finite disjunction are easy.
W
~ is in the
Countable disjunction step: Let ψ = k ψk , and suppose f
~ )KP = Jψk (f
~ )KQ holds for each k ∈ N. Let
reduction of K and that Jψk (f
W
Q
P
′
′
′
~ )K for each k ∈ N, and
~ )K = Jψ (f
ψk = n≤k ψn . Then Jψk (f
k
~ )KQ = Jψ(f
~ )KQ .
~ )KP = lim Jψ ′ (f
~ )KP = lim Jψ ′ (f
Jψ(f
k
k
k→∞
k→∞
Existential quantifier step: Let ψ(~u) = (∃v)θ(~u, v) and suppose that
~ g)KP = Jθ(f
~ , g)KQ for all f
~ , g in the reduction of K. Since the reJθ(f,
duction of K is dense in the sort K parts of both P and Q, and the functions
Jθ(·)KP and Jθ(·)KQ are both Lipschitz continuous with bound 1 by Fact 2.4,
~ )KP = Jψ(f
~ )KQ . This completes the induction.
it follows that Jψ(f
Every event in P has the same measure in V as in V [G]. In particular,
for the sentence ϕ, the measure of JϕKP is the same in V as in V [G]. We
have
V |= µN (JϕK) = µ(JϕKP )
and
V [G] |= µN (JϕK) = µ(JϕKQ ) = µ(JϕKP ).
Therefore µN (JϕK) has the same value in V as in V [G].
Lemma 3.5 can also be proved by using the continuous analogue of the
infinitary logic Lω1 ω . Lemma 5.18 in the paper [EV] shows that for any
metric structure P and continuous infinitary sentence Θ in V , the value of
Θ in P computed in V is the same as the value computed in V [G]. Using
Fact 2.4, one can find a continuous infinitary sentence Θ that has the same
value as µ(JθKP ) in any complete separable randomization P, and then use
Lemma 5.18 in [EV] to get Lemma 3.5.
Lemma 3.6. In any countable transitive model V of ZFC0 , it is true that
every scattered sentence has few separable randomizations.
Proof. By the result of Solovay and Tennenbaum, there is a countable transitive model V [G] of ZFC0 with the same ordinals as V such that V ⊆ V [G]
and Martin’s Axiom for ℵ1 holds in V [G]. Suppose that in V it is true that
ϕ is a scattered sentence, N is a countable randomization with a coding,
and µN (JϕK) = 1.
We now work in V [G], and prove the statement (S) of Lemma 3.4. The
property of being a scattered sentence is Π12 , so by the Shoenfield absoluteness theorem, ϕ is a still scattered sentence. By Lemma 3.5, N is still
a countable randomization with µN (JϕK) = 1. So the completion of N is a
complete separable randomization of ϕ. By Fact 2.10 and Martin’s axiom, ϕ
SCATTERED SENTENCES HAVE FEW SEPARABLE RANDOMIZATIONS
9
has few separable randomizations. By Fact 2.8, there exists a Scott sentence
δ such that µN (JδK) > 0, so (S) holds.
By Lemma 3.4 and the Shoenfield absoluteness theorem (or even the
weaker Mostowski absoluteness theorem), (S) also holds in V . So by Fact
2.8, it is true in V that ϕ has few separable randomizations.
Proof. (Proof of Theorem 3.1) The following argument is well-known, and is
included for completeness. Let η be the sentence in the vocabulary of ZFC
that says that every scattered sentence has few separable randomizations.
Assume ¬η. By the reflection theorem, ZFC0 ∪ {¬η} has a transitive model.
By the downward Löwenheim-Skolem theorem and the Mostowski collapsing
lemma, ZFC0 ∪ {¬η} has a countable transitive model. This contradicts
Lemma 3.6, so η holds.
References
[BBHU] Itaı̈ Ben Yaacov, Alexander Berenstein, C. Ward Henson and
Alexander Usvyatsov. Model Theory for Metric Structures. In Model Theory with Applications to Algebra and Analysis, vol. 2, London Math. Society Lecture Note Series, vol. 350 (2008), 315-427.
[EV] Christopher Eagle and Alessandro Vignati. Saturation and Elementary Equivalence of C ∗ -Algebras. arXiv:1406.4875v4 (2015).
[J] Thomas Jech. Set Theory. Springer-Verlag 2003.
[K1] H. Jerome Keisler. Model Theory for Infinitary Logic. NorthHolland 1971.
[K2] H. Jerome Keisler. Randomizations of Scattered Sentences. To appear in the forthcoming volume “Beyond First Order Model Theory”, edited
by Jose Iovino, CRC Press. Also available online at math.wisc.edu/∼keisler.
University of Wisconsin-Madison, Department of Mathematics, Madison,
WI 53706-1388
E-mail address: [email protected]
URL: www.math.wisc.edu/~andrews
E-mail address: [email protected]
URL: www.math.wisc.edu/~keisler
University of California, Irvine, Department of Mathematics, Irvine, CA,
92697-3875
E-mail address: [email protected]
URL: www.math.uci.edu/~isaac
University of Illinois at Chicago, Department of Mathematics, Statistics,
and Computer Science, Science and Engineering Offices (M/C 249), 851 S.
Morgan St., Chicago, IL 60607-7045, USA
E-mail address: [email protected]
URL: www.math.uic.edu/~marker
E-mail address: [email protected]
URL: www.math.uic.edu/~shac